Trajectory of a rigid body with two wheel with different speed

[Rossi88]

Hi,

first sorry for my bad English, I am more skilled to read than to write in English
I don't know if this is the right section for this topic.

The problem is described in the title of the topic.
My guess is that each of the two wheels goes on with circular trajectory around a same center. There isn't skidding of the wheels.
If the distance between the wheel is D, the speed of the slowest wheel is v1 and the speed of the other wheel is v2 (v1 and v2 is the linear speed), then:

R1=v1*d/(v2-v1)

is the distance between the centre and the slowest wheel, and of course the distance between the centre and the other wheel is

R2=v1*D/(v2-v1)+D=v2*D/(v2-v1)


is my guess correct?
My guess is based on time that each wheel needs to do the circle, and this time needs to be equal for the two wheel.
It seems to be correct, if one wheel is stopped (for example v1=0) the other does a circle around it, infact for the above formulas:
R1=0
R2=D
if the speed of the two wheel are the same
R1->inf
R2->inf

There is someone that knows an other demonstration more rigorous of this?
I attended a mechanical course some year ago, with euler's formula, lagrange and other but I don't remember much and I don't know how to determine the trajectory of a rigid body with two point with different speed.

A special thank to anyone that will help me.

 

[elegysix]

Your equations look good to me, even letting v1=-v2, means R1 and R2 = D/2, which looks good.

 

[Rossi88]

Your equations look good to me, even letting v1=-v2, means R1 and R2 = D/2, which looks good.

your observation is interesting, and is a further proof of the goodness of the equations.
Thank you.

if someone other have a demonstration I will be grateful to him

 

[elegysix]

I think if you took a toy car and put two different size wheels on either side, It would trace the outside of a circle depending on the initial velocity you give it. As long as the wheels are on the same axle that is.

 

[PJC01]

Hello,

Your guess is correct. The trajectory of a rigid body with two wheels moving at different speeds can be determined using the concept of relative motion. Each wheel will have its own circular trajectory around a common center, with the radius of each circle determined by the speeds of the wheels and the distance between them. The center of the circles will also move along a circular trajectory, with its radius determined by the difference in speeds between the two wheels.

To demonstrate this more rigorously, you can use vector mathematics and kinematics equations to analyze the motion of the rigid body. This would involve breaking down the motion of the body into components and using equations such as velocity and acceleration to determine the trajectory of each wheel and the center of the body. You can also use concepts from mechanics such as angular velocity and torque to analyze the motion of the body.

I hope this helps. Keep exploring and learning about the principles of motion and mechanics. Best of luck in your studies!